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February 5, 2010

Sheaves Do Not Belong to Algebraic Geometry

Posted by Tom Leinster

…and here’s a proof.

They are, of course, very useful in algebraic geometry (as is the
equals sign). Also, human beings discovered them while developing algebraic
geometry, which is why many of them still make the association.

But as we’ll see, sheaves are an inevitable consequence of general
ideas that have nothing to do with algebraic geometry.
In fact, sheaves (and various related notions) arise automatically
from two completely general categorical constructions, together with
one almost imperceptibly small topological observation.

Before I give you the proof, let me make clear that it isn’t due
to me. I don’t know who it is due to — I’ve never
seen it in print — but I suspect it was known before I
was even born. (Update: see Joachim Kock’s comment for a reference.) People who I’ve told this argument to seem to like it,
so I wrote it up in a little note
a few years ago; then a recent conversation reminded me of it, so I
thought I’d air it here.

First categorical construction Let A\mathbf{A} be a
small category, E\mathbf{E} a category with small colimits, and
J:A→EJ: \mathbf{A} \to \mathbf{E} any functor. Then there is an induced
adjunction
SetAopHom(J,−)⟵⟶−⊗JE.
\mathbf{Set}^{\mathbf{A}^{op}}
\begin{aligned}
\stackrel{\displaystyle\stackrel{\displaystyle - \otimes J}{\longrightarrow}}{\stackrel{\longleftarrow}{Hom(J, -)}}
\end{aligned}
\mathbf{E}.
The right adjoint Hom(J,−)Hom(J, -) is defined by
(Hom(J,E))(A)=Hom(J(A),E)
(Hom(J, E))(A)
=
Hom (J(A), E)
(E∈EE \in \mathbf{E}, A∈AA \in \mathbf{A}). The left adjoint −⊗J- \otimes J is defined by the adjointness, and can be described as a certain coend or colimit.

Example: if J:Δ→TopJ: \Delta \to \mathbf{Top} is the standard simplex
functor then Hom(J,−)Hom(J, -) is the singular simplicial set functor and
−⊗J- \otimes J is geometric realization.

Second categorical construction Any adjunction restricts
canonically to an equivalence between full subcategories.

Precisely, let
CG⟵⟶FD
\mathbf{C}
\begin{aligned}
\stackrel{\displaystyle
\stackrel{F}{\displaystyle\longrightarrow}}{
\stackrel{\longleftarrow}{G}}
\end{aligned}
\mathbf{D}
be an adjunction (FF left adjoint to GG), with unit η:1→GF\eta: 1 \to G F
and counit ε:FG→1\varepsilon: F G \to 1. Let C¯\bar{\mathbf{C}} be the full
subcategory of C\mathbf{C} consisting of those objects CC for which ηC:C→GF(C)\eta_C: C \to G F(C) is an isomorphism, and
dually D¯\bar{\mathbf{D}}. Then the adjunction (F,G,η,ε)(F, G, \eta, \varepsilon)
restricts to an equivalence between C¯\bar{\mathbf{C}} and D¯\bar{\mathbf{D}}.

Almost imperceptibly small topological observation Any
open subset of a topological space can be regarded as a space in
its own right, and when one open set is contained in another, there is
an induced inclusion of spaces.

Precisely, let SS be a topological space. Write O(S)\mathbf{O}(S)
for the poset of open subsets of SS, regarded as a category (in which
each hom-set has at most one element). Write Top/S\mathbf{Top}/S for the
category of spaces over SS: objects are continuous maps into SS, and
maps are commutative triangles. Then there is a canonical functor
J:O(S)→Top/S,
J: \mathbf{O}(S) \to \mathbf{Top}/S,
sending an open set UU to the inclusion U↪SU \hookrightarrow S.

Punchline Fix a topological space SS. The category
Top/S\mathbf{Top}/S has small colimits, since Top\mathbf{Top} does.

Applying the first categorical construction to the functor JJ just defined produces an adjunction
(presheavesonS)=SetO(S)op⟵⟶Top/S=(spacesoverS).
(presheaves on S)
=
\mathbf{Set}^{\mathbf{O}(S)^{op}}
\begin{aligned}
\stackrel{\displaystyle\longrightarrow}{\longleftarrow}
\end{aligned}
\mathbf{Top}/S
=
(spaces over S).
The two functors here are the ones you’d guess.

Applying the second construction now gives an equivalence of
categories
(sheavesonS)=Sh(S)⟵⟶Et(S)=(étalespacesoverS).
(sheaves on S)
=
\mathbf{Sh}(S)
\begin{aligned}
\stackrel{\displaystyle\longrightarrow}{\longleftarrow}
\end{aligned}
\mathbf{Et}(S)
=
(&eacute;tale spaces over S).
This can be interpreted as the definition of sheaf, étale
space, etc., or as a theorem, according to taste.

Going right and then left in the adjunction gives the associated
sheaf, or sheafification, of a presheaf. Going left and then right
gives the ‘étalification’ of a space over SS.

Posted at February 5, 2010 9:50 PM UTC

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Re: Sheaves Do Not Belong to Algebraic Geometry

I wrote earlier:

It is probably a sign of something that I never thought that sheaves inherently belonged to algebraic geometry.

In contrast, I did think that stacks originally belonged to algebraic geometry. Of course, these are categorified sheaves, hence really more general. But I did think that the algebraic geometers developed them first and furthest.

I write ‘did’ only because now I expect somebody to tell me otherwise. But perhaps I was right.

Re: Sheaves Do Not Belong to Algebraic Geometry

Hi Tom,

To find these constructions together in print, one possibility
is Mac Lane / Moerdijk, Sheaves in geometry and logic, Ch. 2
(p.90–91), with reference to Chapter 1 for the first general
categorical construction.

But you put it much more forcefully! It is beautiful.

I am not sure I agree with the conclusion regarding algebraic
geometry, though :-) If a conclusion is to be drawn, I would
rather put something like “Algebraic Geometry is more than
what you think!”

Re: Sheaves Do Not Belong to Algebraic Geometry

Thanks, Joachim. You’re right: M&M’s account of this is pretty close to what I wrote, much closer than I remembered. Pages 90-91 of my dog-eared copy have my pencilled annotations on them, so I clearly did read that part.

As for what algebraic geometry is, well, that’s a subjective matter. (But your comment does remind me a little of something that Jim Stasheff just said.) I guess I was taught that algebraic geometry was, at its heart, the study of solving polynomials. On the other hand, I never quite believed my teachers — I couldn’t believe that all that cool Grothendieck stuff (which I knew zero about at the time) could possibly be about something so unglamorous.

Re: Sheaves Do Not Belong to Algebraic Geometry

Much of “that cool Grothendieck stuff” was specifically about counting solutions of finitely many polynomial equations over a finite field when they define a smooth projective variety (glamour is clearly in the eyes of the beholder). It always seems to me to be a dreadful misunderstanding to imagine that Grothendieck was interested in building towers of abstractions just for the sake of it. My own vague impression from what I’ve read of “Récoltes et semailles” and other writings of his (e.g., the Grothendieck-Serre letters) is that he would agree with my remark, taking into account that a genius like him doesn’t have the same definition of abstract and concrete as common mortals… (For instance, it is striking how so many of the fundamental properties of étale cohomology turn out to hinge, in final analysis, on seemingly boring old facts of algebraic number theory.)

As for the history of sheaves, it might also be interesting to point out how different Leray’s original definition was: it was based on closed subsets instead of open sets (see Chern’s review of Leray’s main paper). This was re-interpreted and redefined in the late 40’s, in particular during the Cartan seminars. Cartan himself used sheaves to prove “Theorems A and B” in complex analytic geometry, but also used them (before) to re-interpret and extend a result of Oka from the same period (the coherence of the relation sheaf between finitely many holomorphic functions). Interestingly, Oka had used a rough definition of a special case of sheaves (corresponding more or less to ideal subsheaves of the structure sheaf of a complex analytic variety), which he called “idéal holomorphe de domaines indéterminés”. Before these works, very little was known concerning complex analysis (or geometry) in several variables – although there are similarities with algebraic geometry, many facts are much harder (e.g., the coherence of the ideal sheaf of a subvariety is a tautology in algebraic geometry, but is quite hard in the complex setting – this was one of Cartan’s first results beyond Oka’s).

The earliest reference I found in Numdam are here (from a lecture of April 9, 1951, where Cartan states that this is a new take on the 1948/49 definition of sheaf, which I can’t find; Cartan works in the context of a general regular topological space in this work) and here (which contains the results about coherence mentioned above).

As already mentioned by others, sheaves and (Cech, if I remember right) sheaf cohomology were present in Serre’s “Faisceaux algébriques cohérents”, which is a few years pre-schemes; so they were absolutely already at the heart of algebraic and analytic geometry when Grothendieck started working on these topics.

Re: Sheaves Do Not Belong to Algebraic Geometry

Sheaves were developed initially by Leray (although they seem to be traced by Wikipedia back further). He was interested in PDEs rather than anything algebraico-geometric. That was pre-1945. Cartan and Weil used them for a proof of de Rham’s theorem, 1947, then Serre introduced them into Algebraic Geometry only in 1954. The sheaf idea is an extension of analytic continuation. Perhaps therefore sheaf theory is a daughter of analysis, which was mistakenly disowned by its parents and taken under the wing of alg. geom.

Hopefully we can give sheaves back to PDEs and diff. geom. with all current work on approaches to smooth spaces etc. Convincing the analysts that they wrongly disinherited their daughter will be hard though!

On another point, no subject `owns’ sheaf theory, and Joachim’s subversive work for the expansionist Empire of Algebraic Geometry may be a bit like the claims of one world power, A, to dominate a small country, B, essentially because one of A’s historic emperors came from a third small country, C, that conquered B. (I am reading a book on Kublai Khan at the moment if you want to unravel those slightly cryptic comments.)

Re: Sheaves Do Not Belong to Algebraic Geometry

Thanks, Tim. My history is lousy. I remembered that it was Leray who came up with sheaves, but I didn’t know or remember that it was in the context of PDEs.

Nevertheless, I’m convinced that most mathematicians (or at least, those who’ve heard of sheaves) do associate them with algebraic geometry, and that a significant number think that, in some sense, they “belong” to alg geom.

Would you say that it was algebraic geometry that made sheaves famous?

Re: Sheaves Do Not Belong to Algebraic Geometry

The context is confused. I am only using a bit of knowledge other than web sources, such as the Wikipedia article and the St. Andrews’ History one.

http://www-history.mcs.st-andrews.ac.uk/Biographies/Leray.html

He seems to have avoided PDEs as he was in a POW camp and feared that he might be forced to do engineering or similar work for the German authorities. He used his expertise turning towards topology. I note the important quote

… algebraic topology should not only study the topology of a space, i.e. algebraic objects attached to a space, invariant under homomorphisms, but also the topology of a representation (continuous map), i.e. topological invariants of a similar nature for continuous maps.

(This is interesting as another precursor of a categorical viewpoint.)

Was his own motivation for introducing sheaves and spectral sequences topological or analytic? It is not clear.

Sheaves were used in Spanier’s Algebraic Topology book and Hirzebruck’s work, in the 1950s and 1960s, but it is almost certainly EGA and the SGAs and hence Grothendieck, who pushed them to the forefront in Algebraic Geometry, and his prestige has been transfered to them there.

I am not a great one for subdividing mathematics into somewhat arbitrary areas, and tend to emphasise links rather than differences. That is why I think category theory is at the same time very central and often politically marginalised, which route I will turn from as it goes away from the thread! The subdivisions are largely arbitrary.

I have a somewhat related point. Quillen was working in deformation theory in geometry when he started the work on Model Categories, so what area do they fit into? Nowadays people would probably say it was Algebraic Topology, or homotopy theory. The question may be a non-question in some sense.

Re: Sheaves Do Not Belong to Algebraic Geometry

Tim Porter makes some nice observations and some remarks I find myself nodding along to, but unable to add much coherent to. I thought I’d just point out, for those who are interested in some of the history behind Leray’s work during and just after the war, that Haynes Miller has written an interesting account of this:

By the way: I was under the impression, albeit based on little except some browsing in libraries while looking for other things, that there was a period when analysts working in several complex variables took sheaf cohomology (Theorems A, B and all that jazz) pretty seriously and used it.

While this reminds me: has anyone here had cause to look at Finnur Larusson’s recent work advocating a model-categoric/simplicial viewpoint on various parts of holomorphic geometry in higher dimensions? See e.g. this arXiv preprint and various of his sequels.

Re: Sheaves Do Not Belong to Algebraic Geometry

Hold on a sec. Have you pulled a fast one on us again? Your definition of sheaf is not the usual one. Are they obviously equivalent? If so, can you do this for any Grothendieck topology? I had always thought that abstract sheaf theory had exactly one theorem—that the sheafification functor exists. I’d be impressed if you could get around that!

Re: Sheaves Do Not Belong to Algebraic Geometry

I would expect that you’ve just pushed the hard work to proving that this definition of sheaf agrees with the usual one, i.e. a presheaf which behaves in the usual way on covers.

In a sense I agree. If you want to develop the notion of sheaf, starting from the beginning and arriving at the set of basic facts about sheaves that you’re used to, then this won’t save you any work. But I also like the GPS analogy that Urs gives in reply. (Really GPS stands for Gordon, Power and Street, but I also understand this new-fangled usage.) And whenever a definition has several different formulations, people will have subjective feelings about which is the best one.

Re: The abstract definition of sheaves

OK, but what I intended to ask was whether this formal approach allows you to bypass the hard work of constructing the sheafification functor in the usual approach. I would expect that you’ve just pushed the hard work to proving that this definition of sheaf agrees with the usual one, i.e. a presheaf which behaves in the usual way on covers.

Re: The abstract definition of sheaves

whether this formal approach allows you to bypass the hard work

This formal definition allows you to bypass any confusion as to what the notion of sheaf is on an abstract level. This helps you to understand what it is that you need to do in any given situation. Then actually doing it may be hard work, but at least you are doing it efficiently then.

Abstract theory is like a GPS sensor: there is a jungle that you have to cross by foot. It will be hard work to make yourself a way through there. But the general abstract GPS sensor that you carry with you will at least ensure that you go on the straight and direct way through. So as to prevent that on top of all the perils that lurk anyway, you’ll not also lose your way and get lost in the thicket.

constructing the sheafification functor in the usual approach

There is no usual and unusual approach here. All there is is the insight that not only does the sheafification functor turn out to be left exact and left adjoint to the inclusion functor, but that conversely it is entirely characterized by this property.

Re: The abstract definition of sheaves

This formal definition allows you to bypass any confusion as to what the notion of sheaf is on an abstract level.

I think that’s an overly broad statement. I feel that “the notion of sheaf” does not have a unique meaning, and we shouldn’t assert that any one way of thinking about it is the One True Way. This formal definition is one very nice way of thinking about it, which is quick and convenient and gives some good intuition. But the traditional definitions are valid and useful as well, and there are also other valid and useful intuitions.

Re: The abstract definition of sheaves

As I said above, I just wanted to be sure that this point of view doesn’t really cut down on the work you have to do.

That said, I do like this point of view a lot. Somehow I had never appreciated it before. To me, though, it’s particularly worth pointing out that the list of things you need (presheaf, adjoint functor, fully faithful functor, left exact functor) does not include anything like a Grothendieck topology.

Is it true that the point is not so much “sheaves do not belong to algebraic geometry” as much as “sheaves can be defined without sites”?

Re: The abstract definition of sheaves

You don’t need to restrict to sheaves on topological spaces.

What you wrote is of course correct and interesting, but it doesn’t look to me like a generalization of what Tom wrote. I’m not sure if you were saying that it is, but for sheaves on a general site, there is no analogue of the category Top/S\mathbf{Top}/S, as far as I know.

Re: Sheaves Do Not Belong to Algebraic Geometry

I agree with Mike that the statement quoted by Urs is not a generalization of the statement Tom phrased in the introduction. Tom looks at the adjunction between presheaves on X and all spaces over X. Now this adjunction RESTRICTS to an adjunction between a subcategory which can be called sheaves at either side, but Urs’s statement chooses topologically less intuitive side of the “sheaves as presheaves”. By this stage both the info on bigger category Top/X and the geometric look of its subcategory espaces étalé over X is lost. Now Urs explains the generalization of this void-of-usual-spaces-setup corollary (this does not matter at some different levels of questions, but in elementary intuitive foundations it does matter). Now there is a warm up question for Urs: what is REAL higher categorical generalization of étalé spaces (no equivalent categories, description which is analoguous to them, I mean a sort of categorified generalized covering spaces). This is not that simple task. Tim Porter will remember recent thesis at Bangor treating n=2 as one of the issues.

Now, on history. I am so much surprised with the attitude that “sheaves are algebraic geometry”. I was quite knowledgeable about sheaves as undergraduate and studied related algebaric topology a lot then and did not have ANY idea at the time what an algebraic variety is. In our topology seminar which I attended then we used old definitions from 1956 Godement’s excellent book for whom the sheaf is what is here called étalé space and in the Russian edition its total space is called nakryvajushchee prostranstvo, that is covering space; the presheaf point of view is compared of course there in detail (this equivalence of categories proved but not more general adjunction above). Leray, Cartan etc. at the time were interested in analytic continuation like Oka’s problem, Cousin’s problem in cohomology etc. This resulted in EARLY 1950s, that is before Spanier and similar resources irrelevant for early history to lots of results in analytic geometry, Grauert and Remmert taking place on German side and Henri Cartan on French, and of course many others; Cartan’s theorems A and B being some of the famous results. Serre was in the same milleu and transferred in his famous FAC article some of the Cartan seminaire methods to algebraic geometry. Grothendieck was doing tensor products of topological vector spaces at the time. Then he went to Kansas state to do some topology, and there were the breakthrough in sheaf theory, NOT in algebraic geometry. I mean his Kansas report (there are two, the booklet on sheaf theory, and also the official report to the funding agency, where he claims his failure to achieve the goals and then goes on with big discoveries), and his seminar in Spring 1955 where he obtained most of the Tohoku results which were published only 2 years later. Tohoku is still written with main view toward the cohomology of sheaves in topological and equivariant setup. Later AG claimed that the intuition on abelian categories was to isolate the properties he thought of sheaf categories, and that topos theory later was more or less just a later variant of the same initial idea. I mean the characterization of Grothendieck topoi like in Giraud’s theorem is according to him, one of the principal reasons to find the axiomatics of the abelian categories as he phrased it in Tohoku.

In complex analytic geometry according to an intro to a book of Walter Rudin (the one on the complex functions in the unit ball) the sheaves went into second plane only with the discovery of new, quite different, methods of 1970s, which were more effective to then actual questions.

John Baez has recently anounced at the cafe some very recent discovery of Freyd (?) on comparison between abelian and topos case finding a new formulation with closer similarity. I would like to hear a more precise statement and update on this exciting story.

Re: Sheaves Do Not Belong to Algebraic Geometry

what is REAL higher categorical generalization of étalé spaces (no equivalent categories, description which is analoguous to them, I mean a sort of categorified generalized covering spaces). This is not that simple task. Tim Porter will remember recent thesis at Bangor treating n=2 as one of the issues.

I would posit that for n=2 the answer is: 2-covering spaces, as defined in my thesis (see chapter 4). The thesis you refer to, by Richard Lewis, takes a more algebro-combinatorial approach, with crossed modules acting on simplicial sets.

Re: Sheaves Do Not Belong to Algebraic Geometry

Thanks Zoran and David.

Yes. I felt from way back when I started working on 2-stack type things (actually before we at Bangor got the start of the Grothendieck correspondence, (in the sense of letters not a bijection!!!!), but after, by several years, AG’s letter to Breen (note not the later letter to Quillen)), that the 2-type of a space classified its 2-covering spaces, that that was very important, and that if I was to attempt to convince analysts, geometers and others that this was important, there had to be fairly concrete models of the covering spaces. The solution using abstract sheaf theoretic models should be doable (and probably is essentially there in places that I should have read!!!) such as David’s thesis, which I have only browsed, but what about Galois theory, Riemann surface theory, ramifications, complex analytic continuation, etc. How were generalisations of those to be found and then `sold’ to the non-categorical mathematical audience? That needed something that I could not visualise. Something more concrete than just an abstract construction in a Quillen model category, however pretty, is needed. I had seen algebraic topologists in seminars ‘turn off’ if you mentioned anything non-stable, yet what I wanted was very basic, some sort of elementary approach to those ideas.

I felt those questions were important then and still do now. Some of John’s ideas and a lot of the stuff here in the café and in the Lab go a long way towards answering them.

Grothendieck’s letters to Ronnie, and their transformation into Pursuing Stacks made me very pleased!!! But then I had to start learning all the bits of theory, that would be needed to do put his ideas into practice. I am still learning!

Re: Sheaves Do Not Belong to Algebraic Geometry

Here’s another interesting aspect of this. It’s certainly true that any adjunction restricts to an equivalence between the full subcategories consisting of the objects for which the unit or counit (as appropriate) is an isomorphism. However, for many adjunctions, these full subcategories can be very small. For instance, for the free-forgetful adjunction of any algebraic theory (monoids, groups, rings, etc.), these full subcategories are empty.

But the adjunction in question here has a special property: it’s idempotent. That means that the unit ηC:C→GF(C)\eta_C \colon C\to G F (C) is an isomorphism whenever CC is of the form G(D)G(D), and (equivalently!) that the counit εD:FD(D)→D\varepsilon_D\colon F D(D) \to D is an isomorphism whenever DD is of the form F(C)F(C).

In general, you could think of starting with an object CC and trying to get it into one of the full subcategories that are equivalent by successively applying the functors: C,F(C),GF(C),FGF(C),…C, F(C), G F(C), F G F(C), \dots. In general, you’ll never get there, while the adjunction being idempotent means that you get there after only one step. (Off the top of my head, I don’t know any examples where you get there in some finite number of steps >1\gt 1.)

Re: Sheaves Do Not Belong to Algebraic Geometry

Re: Sheaves Do Not Belong to Algebraic Geometry

Hi I know its an awfully long time since this was posted but i just read it and couldn’t help thinking about this. The question being how does one show “slickly” that the adjunction derived in the first place is idempotent? For this i propose the following Lemma:

If this lemma is correct (i think the proof is below), then the situation with sheaves is a corollary since in that case z:𝒪(X)→Top/Xz : \mathcal{O}(X) \rightarrow \text{Top}/X is clearly full since the only arrows of the form z(U)→z(V)z(U) \rightarrow z(V) in Top/X\text{Top}/X must be inclusions.

Here is a proposed proof: Let η\eta be the unit of the adjunction. ηG\eta G is trivially a split monic from the triangle identities, so we are only concerned with showing it is epi, for which it suffices to show that each of the embedded components, of the following form, is a surjection of sets:

… which function acts by mapping the arrows through GG, so an arrow h:zC→Dh \;:\; z C \rightarrow D on the left becomes h*h_{*} on the right.

Conversely, consider any arrow φ:GzC→GD\varphi \;:\; G z C \rightarrow G D on the right. Written out in full, this is a natural transformation φ:Hom𝔼(z−,zC)→Hom𝔼(z−,D)\varphi \;:\; \text{Hom}_{\mathbb{E}}(z -,z C) \rightarrow \text{Hom}_{\mathbb{E}}(z -,D). Draw a naturality square for this with respect to an arbitrary arrow g:C′→Cg \;:\; C' \rightarrow C and chase the unit idzC\text{id}_{z C} around the diagram to learn that φC′:(zg)↦φC(idC)⋅(zg)\varphi_{C'} \;:\; (z g) \mapsto \varphi_{C}(\text{id}_C) \cdot (z g). Provided that zz is full, this allows us to conclude that φ\varphi is of the form h*h_* where h=φC(idC)h = \varphi_C(\text{id}_C) in all cases, i.e. that (ηGD)C(\eta_{G D})_C is surjective as required, and the adjunction is idempotent.

Re: Sheaves Do Not Belong to Algebraic Geometry

I’m afraid I’ve only just sat down to work through this.

I’d love to see a simple criterion for when the adjunction ℂ^→←𝔼\widehat{\mathbb{C}} \stackrel{\leftarrow}{\rightarrow} \mathbb{E} induced by a functor ℂ→𝔼\mathbb{C} \to \mathbb{E} is idempotent, but I don’t think this lemma can be right.

Here’s what appears to be a counterexample. First recall that an adjunction is idempotent if and only if the comonad it induces is idempotent (that is, the multiplication part of the comonad is an isomorphism).

Now consider the inclusion FinSet→SetFinSet \to Set of the full subcategory of finite sets, and form its opposite FinSetop→SetopFinSet^{op} \to Set^{op}, which is certainly full (and faithful). The adjunction induced by this latter functor is an adjunction between SetopSet^{op} and SetFinSetSet^{FinSet}. It induces a comonad on SetopSet^{op}, which of course can be thought of as a monad on SetSet.

However, the ultrafilter monad UU is not idempotent (at least, assuming that SetSet satisfies the axiom of choice). For if UU is idempotent then for each set XX, the multiplication map UUX→UXU U X \to U X is a bijection, or equivalently the unit map ηUX:UX→UUX\eta_{U X} : U X \to U U X is a bijection, and therefore every ultrafilter on UXU X is principal. But if we take XX to be infinite, then UXU X is also infinite, so (by choice) UXU X admits a non-principal ultrafilter — a contradiction.

What’s wrong with hypnocat’s proof? Well, it seems to me that the codomain of (ηGD)C(\eta_{G D})_C (shown in the displayed equation) is wrong. (Here C∈ℂC \in \mathbb{C} and D∈𝔼D \in \mathbb{E}.) I think that display should read

Re: Sheaves Do Not Belong to Algebraic Geometry

This is a really cool post! I was recently reading about nerves and realization, and I realized that in fact this construction described in your blog is a special case this: the right adjoint here is the nerve, and the left adjoint is the realization. So taking the sheaf of local sections of an etale space is just taking the nerve, and constructing the etale space for a sheaf is geometric realization!